Graphical element rooftop reconstruction in digital map

ABSTRACT

A client device receives a first map tile, a second map tile, and map terrain data from a mapping system, the first and second map tiles together including map feature having a geometric base with a height value, the geometric base represented by a set of vertices split across the first and second map tiles. The client device identifies edges of the geometric base that intersect a tile border between the first and second map tiles. The client device determines a set of sample points based on the identified edges and determines a particular sample elevation value corresponding to a sample point in the set. The client device renders the map feature based on the particular sample elevation value and displays the rendering of the map feature.

FIELD OF ART

This description relates to rendering digital maps, and particularly togenerating rooftops on digital maps.

BACKGROUND

Digital maps are widely used today across myriad domains and for variouspurposes, from navigation, to ride sharing, to undergirding video games,among many others. While a standalone digital map computer programapplication may include data and processes requisite for its functionalpurpose, other digital map computer program applications exposefunctionality involving digital maps by communicating with a physicallyand/or logically separate mapping system (e.g., a digital map server),such as through employment of an Application Programming Interface (API)of the separate mapping system to access its data and/or processes.

Some conventional mapping systems render digital maps by stitchingtogether map tiles each depicting a respective portion of a digital map.The conventional mapping system may include multiple sets of map tilesat a variety of “zoom levels,” i.e., levels of granularity (e.g., a setof map tiles dividing the earth into sixty-four regions, and another setof map tiles dividing the earth into tens of thousands of regions),where the set of map tiles employed by the conventional mapping systemfor a given digital map depends upon a zoom level of the digital map(e.g., the extent of an area captured within a viewport of the digitalmap at a given time).

Each map tile is associated with a set of map features that may bedigital representations of geographic features of the geographic regionrepresented by the map tile, such as roads, bodies of water, publicparks, buildings, or so on, as well as metadata corresponding togeographic features, such as textual labels, icons, symbols, or so on.The conventional mapping system can include graphical elementscorresponding to map features that can be rendered in order to visuallyrepresent respective map features in a digital map.

Different map tiles at different zoom levels but representing the samegeographic region may include different map features for the geographicregion. For example, a first map tile representing 1,000 square milesmay include as a map feature a label for a particular city at aparticular geographic location, while a second map tile representing 10square miles may include the label for the particular city at theparticular geographic location, as well as a variety of map featuresrepresenting roads, buildings, parks, and so on, each representing ageographic feature of the city.

The conventional mapping system loads a set of map tiles to use togenerate a digital map, either from local storage, or by querying aseparate mapping system. As the zoom level of the digital map changes(e.g., in response to user input), the conventional mapping system mayresponsively load map tiles corresponding to the new zoom level on thefly, and dynamically render newly loaded map tiles as they are loadedfrom local storage and/or received from a separate mapping system. Someconventional mapping systems wait until all map tiles are loaded, oralternatively a separate mapping system loads all relevant map tilesbefore sending them to the conventional mapping system, but this canlead to significant time delays that reduce the functionality of thedigital map, especially for purposes that are time-sensitive, such asvehicle navigation or video game play.

Graphical elements representing map features may cumulatively includeenough data that rendering them requires significant processingcapabilities, and the rendering process may be slow enough that itnegatively impacts the functionality of the digital map. Renderingthree-dimensional graphical elements in particular can becomputationally complex and expensive in terms of computing resources.For example, conventional techniques to generate a three-dimensionalmodel of a building can involve calculating an elevation value for eachpoint on the building's footprint, determining a building base elevationbased on the elevation values, then drawing a roof a certain distanceabove the points of the building's footprint according to a height valueof the map feature and the determined building base elevation. This andother techniques typically create a computational performance bottleneckdue to the significant computing resources involved in executing suchtechniques for the (often expansive) set of buildings within theviewport of the digital map at any given time, which can change quicklyin response to user manipulation of the viewport (e.g., scaling,panning, tilting, rotating, or other digital map operations), thusnecessitating new roof computations and further exacerbating theproblem.

Regardless of when, or from where, the conventional mapping systemacquires map tiles to generate a digital map, additional problems arisewith rendering graphical features falling on the border between two maptiles. This is especially true when computing the roof of a buildingthat lay across two map tiles, in no small part due to the commonsituation where the conventional mapping system has not yet loaded allmap tiles currently needed to render the digital map. As such, theconventional mapping system may not have the elevation values for asubset of the points on the building's footprint that are on an unloadedmap tile. Also, the conventional mapping system may not have anassociation between two subsets of points of the same building that areon two different map tiles, and thus not have any indication that theybelong to the same building and should be rendered together. Either casecan lead to a miscalculation of the building base elevation, and/or leadto the conventional mapping system treating each subset of points of thebuilding's footprint as separate buildings.

The conventional mapping system may use a first subset of points of thebuilding's footprint on a first map tile to generate a first buildingbase elevation and render a first rooftop based on the first buildingbase elevation. The conventional mapping system then uses a secondsubset of points of the building's footprint from a second map tile togenerate a second building base elevation and render a second rooftopbased on the second building base elevation. If the first and secondbuilding base elevations are different, this can cause what should berendered as one building graphical element to instead appear to be twodifferent structures, with a roof that sharply changes in height at themap tile border. This not only doubles the computational load on theprocessor, but also creates an unrepresentative graphical element thatdoes not accurately visualize the map feature. As such, improved systemsfor rendering digital maps are needed.

SUMMARY

According to one embodiment of a method, a client device receives, froma mapping system, a first map tile representing a first portion of ageographic region. The first map tile is associated with a map featurethat has a geometric base and a height value. The geometric base isrepresented by a set of vertices that includes one or more vertices onthe first map tile. The client device also receives a second map tilerepresenting a second portion of the geographic region. The set ofvertices includes one or more vertices on the second map tile. Theclient device also receives map terrain data including elevation valuesfor points on the first map tile and points on the second map tile. Theclient device identifies, based on the set of vertices, one or moreedges of the geometric base that each connects two vertices of the setof vertices and intersects a tile border between the first map tile andthe second map tile.

Responsive to receiving each of the first map tile, the second map tile,and the terrain data, the client device determines a set of samplepoints for the map feature based on the one or more edges that intersectthe tile border, selects a particular sample elevation value from thedetermined sample elevation values, and determines vertex elevationvalues for respective vertices in the set of vertices.

The client device renders the first map tile and the second map tilewith the map feature using the vertex elevation values. The map featureis rendered by the client device with a top surface graphical componenthaving an elevation corresponding to a sum of the particular sampleelevation value and the height value of the map feature.

The client device displays a graphical element of the map featurecomprising the top surface graphical component on an electronic displayof the client device. For example, if the map feature is a building, theclient device displays a digital map including the building, asrepresented by a graphical element based on the geometric base andtopped by the top surface graphical component at the elevationdetermined as a sum of the particular sample elevation value and heightvalue.

In this manner, the client device not only lessens the computationalload on the processor used to generate the top surface graphicalcomponent of the building, but also creates a more representativegraphical element that more accurately visualizes the map feature withinthe digital map.

In an embodiment, the client device determines the set of sample pointsby determining, based on the identified one or more edges, a borderintersection span for the map feature, the border intersection spanrepresenting a projection of the one or more edges onto the border, andselecting one or more end points of the border intersection span forinclusion in the set of sample points.

In an embodiment, the client device determines the set of sample pointsby determining a vertex of the set of vertices having a maximum distancefrom the border relative to the other vertices of the set of verticesand including the vertex as a sample point in the set of sample points.

In an embodiment, the client device receives the one or more of thefirst map tile, the second map tile, and the terrain dataasynchronously.

In an embodiment, the client device determines the particular sampleelevation value by selecting the maximum sample elevation value from thesample elevation values for the set of sample points.

In an embodiment, after rendering the first and second map tiles, theclient device receives updated map data including updates to one or moreof the first map tile, the second map tile, the map feature, and the mapterrain data. The client device determines an updated sample elevationvalue for an updated sample point of the set of sample points using theupdated map data. The client device determining updated vertex elevationvalues for the set of vertices using the updated sample elevation value.The client device re-renders the first and second map tiles with the mapfeature using the updated vertex elevation values. The map feature isrendered with the top surface graphical component having an elevationcorresponding to a sum of the updated sample elevation value and theheight value. The client device displays the re-rendered first andsecond map tiles on the electronic display of the client device.

In an embodiment, the client device determines the vertex elevationvalues by, responsive to receiving the map terrain data, evaluating eachmap tile of the first and second map tiles for map features intersectingthe tile border of the first map tile and second map tile. This caninvolve the client device identifying a first portion of the map featureon the first map tile intersecting the tile border between the first andsecond map tiles. The client device then compares the first portion ofthe map feature to one or more map features of the second map tile. Theone or more map features of the second map tile include a second portionof the map feature. The client device then determines, based on thecomparison, that that first portion of the map feature matches thesecond portion of the map feature of the second map tile. The clientdevice, responsive to determining that the first and second portionsmatch, determines the vertex elevation values for the set of verticesbased on the matching first and second portions of the map feature.Depending upon the embodiment, this may further involve, in a case wherethe first map tile is received before the second map tile, and beforereceiving the second map tile, responsive to determining that one ormore vertices of the set of vertices are not on the first map tile,rendering, by the client device, the first map tile without the mapfeature. Determining the vertex elevation values may, in an embodiment,involve, before evaluating each map tile of the first and second maptiles for map features intersecting the tile borders of the first andsecond map tiles, ordering, by the client device, one or more mapfeatures of the first map tile that intersect a first tile border of thefirst map tile according to a point of intersection of the one or mapfeatures on the first tile border. The client device then orders one ormore map features of the second map tile that intersect a second tileborder of the second map tile according to a point of intersection ofthe one or map features on the second tile border.

In an embodiment, the client device stores the vertex elevation valuesin a vertex shader, and the first and second map tiles are rendered withthe map feature by the GPU of the client device using the vertex shader.

In another embodiment, a non-transitory computer-readable storage mediumstores instructions that, when executed by a computing device, executesone or more of the above-described methods.

In yet another embodiment, a computer system includes a processor and anon-transitory computer-readable storage medium that stores instructionsfor executing one or more of the above-described methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a computer environment in which the techniquesdescribed may be practiced, according to one embodiment.

FIG. 2A is a block diagram illustrating a mapping system, according toone embodiment.

FIG. 2B is a block diagram illustrating a client device, according toone embodiment.

FIG. 3 illustrates a three-dimensional digital map including a buildinggraphical element with a misaligned roof, according to an embodiment.

FIG. 4A illustrates a top-down topographic view of calculating a samplepoint for rooftop reconstruction in a digital map, according to anembodiment.

FIG. 4B illustrates a side view of rooftop reconstruction in a digitalmap, according to one embodiment.

FIG. 5 illustrates a top-down topographic view of calculating a samplepoint for efficient rooftop rendering in a digital map, according to anembodiment.

FIG. 6 is a flowchart illustrating a process for rooftop reconstruction,according to one embodiment.

FIG. 7 is a flowchart illustrating a process for efficient rooftoprendering, according to one embodiment.

FIG. 8 is a block diagram that illustrates a computer system upon whichembodiments of components of the computing environment depicted in FIG.1 may be implemented, according to an embodiment.

The figures depict embodiments of the present invention for purposes ofillustration only. One skilled in the art will readily recognize fromthe following description that alternative embodiments of the structuresand methods illustrated herein may be employed without departing fromthe principles of the invention described herein.

DETAILED DESCRIPTION

I. System Environment

FIG. 1 illustrates a computer environment 100 in which the techniquesdescribed may be practiced, according to one embodiment. In theembodiment shown, the computing environment 100 includes a mappingsystem 110 that provides mapping services and a client device 120 thatcommunicates with the mapping system 110 to receive map data via anetwork 130. In different embodiments, the computing environment 100 andits components may include different or additional elements than thoseillustrated in FIGS. 1-2 . Furthermore, the functionality may bedistributed among the elements in a different manner than described. Thecomputer environment 100 comprises components that are implemented atleast partially by hardware at one or more computing devices, such asone or more hardware processors executing stored program instructionsstored in one or more memories for performing the functions that aredescribed herein. In other words, all functions described herein areintended to indicate operations that are performed using programming ina special-purpose computer or general-purpose computer, in variousembodiments. The components of FIG. 1 are now described in more detail.

The mapping system 110 provides mapping services to client devices. Inembodiments, the mapping system 110 provides map data to client devicesfor a variety of purposes, such as to display digital maps for purposesof navigation, ride sharing, or video games (e.g., a mixed realitygame). In particular, the mapping system 110 provides the client device120 with map tiles representing geographic regions, which can be used torender digital maps at various scales or resolutions, such as bystitching together a set of map tiles. For example, the client 120 maysend a request to the mapping system 110 for map tiles at a particularzoom level for a particular geographic region, and the mapping system110 responds to the query by sending the client device 120 a respectiveset of map tiles, which is duly received by the client device 120.

Each map tile is associated with a set of map features that may bedigital representations of geographic features of the geographic regionrepresented by the map tile, such as roads, bodies of water, publicparks, buildings, or so on, and/or may be metadata corresponding togeographic features, such as textual labels, icons, symbols, or so on.The map features of a map tile are rendered by the client device 120using corresponding vector data. The vector data of a map tile includesgeometric information representing the geographic features representedby the map features, such as points (e.g., vertices), lines (e.g.,edges), and shapes (e.g., polygons), which may be associated withparticular geographic coordinates and/or tile coordinates (e.g.,coordinates indicating a position of the map feature within the maptile). For example, a map feature may be represented by a set ofvertices and edges among those vertices that collectively describe theshape of a graphical element that visualizes the map feature. The set ofvertices and edges may be divided into graphical components, such as atop surface graphical component (e.g., a roof).

The client device 120 processes the geographic information and labels ofa map tile to render the map tile, which can be combined with otherrendered map tiles to render an overall digital map. By renderingindividual map tiles representing portions of an overall digital map, asopposed to rendering the entire digital map altogether, the clientdevice 120 provides for efficient rendering of digital maps that candynamically adapt to various scenarios (e.g., map scales).

The mapping system 110 may be comprised of any computing device,including but not limited to: servers, racks, workstations, personalcomputers, general purpose computers, laptops, Internet appliances,wireless devices, wired devices, multi-processor systems,mini-computers, virtual computing instances, and the like. Although FIG.1 shows a single element, the mapping system 110 broadly represents oneor multiple computing devices, such as a server cluster running adistributed computer program, where individual servers of the clustermay be located in one or more discrete physical locations.

The client device 120 communicates with the mapping system 110 in orderto receive map data. The client device 120 may use the map data for avariety of purposes, such as rendering and displaying a digital map, orproviding navigation services. In embodiments the client device 120requests a set of map tiles from the mapping system 110 for rendering adigital map on a display of the client device 120. For instance, theclient device 120 may request a set of map tiles for rendering a digitalmap representing a particular geographic region at one or more scales orresolutions. In this case, the client device 120 may receive the set ofmap tile from the mapping system 110 and stitch together the set of maptiles to render the digital map. As described in further detail below,stitching together the set of map tiles can involve determininggraphical components of graphical elements constructed from vector datato visualize map features in the digital map. In particular, the clientdevice 120 may determine top surface graphical components (e.g.,rooftops) for graphical elements corresponding to map features thatoverlap multiple map tiles.

The client device 120 may be any suitable computing device, such as alaptop computer, hand-held computer, wearable computer, cellular ormobile phone, portable digital assistant (PDA), kiosk, or tabletcomputer. Although a single client device is depicted in FIG. 1 , anynumber of client devices may be present. The client device 120 mayinclude a GPS receiver that receives signals describing a geographicposition of the client device 120 from GPS satellites (e.g., longitudeand latitude). The client device 120 includes an electronic display onwhich digital maps may be presented.

In some embodiments, the client device 120 executes a client mapapplication associated with mapping system 110, such as software thatdisplays, uses, supports, or otherwise provides electronic mappingfunctionality as part of the application or software. The client mapapplication may be configured for a variety of functions involving mapdata, such as navigation, transportation, augmented reality, productdelivery, etc. In one embodiment, the client map application obtainselectronic mapping functions through an integrated software developmentkit (SDK) or an application programming interface (API) associated withthe mapping system 110. In this case, the SDK or API may implementfunctional calls, callbacks, methods or other programmatic means forcontacting the server computer to obtain map tiles, map tile labels,layer data, digital elevation models, or other data useful to visuallyrender a digital map as part of the application. As an example, theclient map application may identify a set of map tiles needed to rendera particular geographic region (e.g., defined by a set of geographiccoordinates indicating a current or upcoming viewport of the digital mapat the electronic display of the client device 120) and requestcorresponding map tiles and/or related data (e.g., terrain data such asdigital elevation models) from the mapping system 110.

The network 130 connects the mapping system 110 and the client device120. The network 130 may be any suitable communications network for datatransmission. In an embodiment such as that illustrated in FIG. 1 , thenetwork 130 uses standard communications technologies or protocols andcan include the internet. In another embodiment, the entities use customor dedicated data communications technologies.

II. Exemplary Mapping Systems

FIG. 2A is a block diagram illustrating a mapping system, according toone embodiment. In the embodiment shown, the mapping system 110 includesa map data interface 210 that communicates with external systems ordevices (e.g., the client device 120) to provide mapping services and amap tile coordination module 220 that coordinates the retrieval of mapdata relevant to a request from the client device 120, e.g., a set ofmap tiles for display on the client device 120. The mapping system 110further includes a digital map source data store 230 that stores rawdigital map source data, a digital map data store 240 storing digitalmap data (e.g., derived from the digital map source data), and atelemetry data store 250 storing telemetry data. The components of FIG.2 are now described in more detail.

The map data interface 210 provides map data services to one or morecomputing systems (e.g., the client device 120). In particular, the mapdata interface 210 receives requests for map data describing ageographic region, such as requests for a set of map tiles that can beused to render a digital map representing the geographic region at oneor more scales or resolutions (e.g., by stitching together the set ofmap tiles). The map data interface 210 may request map tiles from themap tile coordination module 220 that correspond to the geographicregion for a particular zoom level. Responsive to receiving therequested map tiles from the map tile coordination module 220, the mapdata interface 210 may provide the map tiles to the requesting system ordevice. The map data interface 210 may receive a request for map datafrom various sources, such as another component of the mapping system110 or from a map application on the client device 120. The map datainterface 210 may provide various interfaces to facilitate communicationbetween the mapping system 110 and other computing systems, such as userinterfaces, application programming interfaces (APIs), SoftwareDevelopment Kits (SDKs), and/or any other suitable interface.

The map tile coordination module 220 interfaces with the digital mapsource data store 230, digital map data store 240, and telemetry datastore 250 to retrieve data relevant to a data request, e.g., a requestfrom a client device 120 ingested by the map data interface 210 andtranslated into a request to the map tile coordination module 220. Themap tile coordination module 220 retrieves the requested data from oneor more data stores. For example, the map tile coordination module 220generates one or more database queries and sends the database queries toone or more respective data stores, such as the digital map source datastore 230, the digital map data store 240, or the telemetry data store250, depending upon the embodiment. The map tile coordination module 220receives data from one or more data stores in response to the one ormore database queries. The map tile coordination module 220 sends thedata to the map data interface 210 for propagation to the client device120.

The digital map source data store 230 stores raw digital map data thatis obtained, downloaded or received from a variety of sources. The rawdigital map data may include satellite images, digital street data,building data, place data or terrain data. Example sources includeNational Aeronautics and Space Administration (NASA), United StatesGeological Survey (USGS), and DigitalGlobe. The digital map source datastore 230 may be updated at any suitable interval, and may be stored forany amount of time. Once obtained or received, raw digital map sourcedata stored in the digital map source data store 230 can be used by themapping system 110 to generate digital map data (e.g., stored in thedigital map data store 240).

The digital map data store 240 stores digital map data including maptiles. The digital map data may be derived from digital map source data,e.g., stored in the digital map source data store 230. In embodiments,the mapping system 110 processes and organizes digital map source dataas a plurality of map tiles with corresponding sets of map labels orother style data used to impose different display styles for renderingthe map tiles. Digital map data stored in the digital map data store 240may be updated at any suitable interval, and may include additionalinformation beyond that derived from digital map source data in the datastore 230. For example, using aggregated telemetry data stored in thetelemetry data store 250, discussed below, various additionalinformation may be stored in association with the map tiles, such astraffic patterns, turn restrictions, detours, common or popular routes,speed limits, new streets, and any other information related to digitalmaps or the use of digital maps.

The telemetry data store 250 stores digital telemetry data that isobtained or received from mobile computing devices via function callsthat are included in a Software Development Kit (SDK) that applicationdevelopers use to integrate and include electronic maps in applications.The telemetry data may include mobile device location information basedon GPS signals. For example, telemetry data may comprise one or moredigitally stored events, in which each event comprises a plurality ofevent attribute values. Telemetry events may include: session start, mapload, map pan, map zoom, map tilt or rotate, location report, speed andheading report, or a visit event including dwell time plus location.Telemetry event attributes may include latitude-longitude values for thethen-current position of the mobile device, a session identifier,instance identifier, application identifier, device data, connectivitydata, view data, and/or timestamp. The telemetry data store 250 maystore aggregated telemetry data that has been processed using userprivacy-preserving operations such as anonymization or chunking,data-cleaning operations such as filtering, or a combination thereof.Anonymization may include removing any data that identifies a specificmobile device or person. Chunking may include segmenting a continuousset of related telemetry data into different segments or chunksrepresenting portions of travel along a route. For example, telemetrydata may be collected during a drive from John's house to John's office.Chunking may break that continuous set of telemetry data into multiplechunks so that, rather than consisting of one continuous trace, John'strip may be stored as a trip from John's house to point A, a separatetrip from point A to point B, and another separate trip from point B toJohn's office. Chunking may also remove or obscure start points, endpoints, or otherwise break telemetry data into any size. Filtering mayremove inconsistent or irregular data, delete traces or trips that lacksufficient data points, or exclude any type or portion of data for anyreason. Once processed, aggregated telemetry data can be stored inassociation with one or more map tiles, e.g., stored in the digital mapdata store 240. Aggregated telemetry data may be stored for any amountof time, such as a day, a week, or more. Aggregated telemetry data maybe further processed or used by various applications or functions asneeded.

The data stores 230, 240, or 250 may span multiple devices located inone or more physical locations. For example, the data stores 230, 240,or 250 may include one or more nodes located at one or more datawarehouses. Furthermore, although the data stores 230, 240, and 250 aredepicted in FIG. 1 as components of the mapping system 110, the datastores 230, 240, or 250 may be located on a separate device or devicesfrom the mapping system 110.

FIG. 2B is a block diagram illustrating a client device 120, accordingto one embodiment. The client device 120 includes a central processingunit (CPU) 260, and in various embodiments includes a graphicalprocessing unit (GPU) 290. In the embodiment shown, the client device120 includes a map tile preparation module 270 that communicates withexternal systems or devices (e.g., the mapping system 110) to requestand receive map data, such as map tiles, terrain data, navigationinstructions, and/or other data related to digital maps. The clientdevice 120 further includes a rendering module 280 that uses map datareceived from the mapping system 110 and/or retrieved from local storage(not illustrated). In an embodiment, the client device 120 stores mapdata locally on the client device 120, e.g., some or all map datareceived from the mapping system 110. Depending upon the embodiment,some or all processes of the map tile preparation module 270 and/orrendering module 280 may be performed solely by the CPU 260. However, insome embodiments, the client device 120 performs functionality of therendering module 280 using the GPU 290, as further detailed below.

The rendering module 280 performs rooftop reconstruction, which is atechnique to generate a top surface graphical component of a graphicalelement visualizing a map feature in cases where the map featureoverlaps multiple map tiles. Using rooftop reconstruction, the renderingmodule 280 enables the client device 120 to reduce the computationalexpense of rendering the digital map, while simultaneously resolvingrendering problems where two partial top surface graphical componentswith different elevations are generated for the map feature, thusvisualizing the map feature with an unrealistic roof.

A situation where the map feature may be rendered as two separatepartial buildings, for example, is when the first map tile is rendered,and then the second map tile is received and then rendered. As the dataof the second map tile was not present when the first map tile wasrendered, the rendering module 280 did not have indication of verticesof the map feature within the second map tile. (In such a case, asdetailed below, a rendering module 280 employing rooftop reconstructiontechniques would regenerate a singular top surface graphical componentfor the building upon loading the second map tile). As a second example,the map tiles may not include data indicating correspondence between twoor more sets of vertices in different map tiles that collectively formthe geometric base of one building. (This is generally the case, asincluding such identifiers in practice significantly increases the sizeof the data, increasing the network bandwidth needed to transmit it, aswell as the computational resources needed to store it and employ it).

The rendering module 280 receives a set of map tiles including a firstmap tile and a second map tile, as well as map terrain data, from themapping system 110 (e.g., by way of the map tile preparation module270). Depending upon the embodiment, the rendering module 280 mayreceive one or more map tiles and/or the map terrain data separately ortogether, synchronously or asynchronously. Each of the first and secondmap tiles represents a different portion of a geographic region (i.e., adifferent area segment of geographic space) than the other. The firstmap tile includes a map feature with a geometric base and a heightvalue. The geometric base is a set of vertices and a set of edgesrepresenting a shape of the map feature. Each vertex in the set ofvertices has a coordinate location within the first map tile. Each edgein the set of edges connects two vertices in the set of vertices. Theheight value indicates a height of a top surface graphical component ofthe map feature above the geometric base (e.g., a height of a roof abovea building's floor).

One or more of the vertices in the set of vertices is located within thearea represented by the first map tile. A separate one or more of thevertices in the set of vertices is located in the second map tile. Assuch, the map feature overlaps the first map tile and the second maptile. In particular, the map feature intersects a tile border betweenthe first map feature and the second map feature. That is, one or moreedges of the set of edges intersects the tile border by connecting apair of vertices with one vertex in the first map tile and the other inthe second map tile.

In an embodiment, the set of vertices is divided into a first subset ofvertices in the first map tile and a second subset of vertices in thesecond map tile, and the rendering module 280 does not have dataindicating the subsets collectively form one map feature's geometricbase (e.g., the map tiles do not include map feature data indicatingwhether vertices of the map feature are on other map tiles). Therendering module 280 determines whether vertices on different map tilesbelong to the same map feature (and thereby identify map features towhich rooftop reconstruction applies) as follows. The rendering module280 identifies map features on each map tile that terminate at the tileborder. The rendering module 280 determines whether any identified mapfeature in the first map tile is within a threshold distance from anidentified map feature in the second map tile (e.g., whether coordinatesof one or more vertices of one identified map feature along the tileborder are less than a threshold distance from coordinates of one ormore vertices of another identified map feature along the tile border).If two identified map features (one on each map tile) are within thethreshold distance, the rendering module 280 identifies them asmatching. Responsive to determining that the first and second identifiedmap features match, the rendering module 280 combines the vertices ofeach of the first and second identified map features (e.g., each subsetof vertices) into one set of vertices for the map feature as a whole.

In an embodiment, if the client device 120 has received the first maptile but not the second map tile, and the client device 120 determinesthat one or more map features on the first map tile terminate at thetile border, the client device 120 does not render the one or more mapfeatures until the second map tile is received and the above process isperformed.

In an embodiment, before evaluating each map tile of the first andsecond map tiles for map features intersecting the tile borders of thefirst and second map tiles, the rendering module 280 orders one or moremap features of the first map tile that intersect the tile borderaccording to respective points of intersection (e.g., in ascending ordescending order). For example, a particular coordinate value mayincrease or decrease along the tile border. The rendering module 280orders map features terminating in the tile border in order by theparticular coordinate value. The rendering module 280 performs a similaroperation for map features in the second map tile that terminate at thetile border. In this manner, the rendering module 280 makes the processof matching partial map features to one another more computationallyefficient by lessening the number of comparisons that need be madebetween identified map features.

The map terrain data is data indicating elevation values at one or morepoints (e.g., coordinate locations) within the digital map, andcorresponds to points on the map tiles. The map terrain data indicates adifference in elevation from a base elevation of the digital map (e.g.,sea level). For example, a first point on the first map tilerepresenting a particular geographic location within the geographic arearepresented by the first map tile may have an elevation value of 1,000to represent that a ground level elevation of 1,000 feet above sea levelis found at the particular geographic location (e.g., as a location upona hillside). In some embodiments, elevation values can be negative,e.g., to indicate elevations that are below sea level. In an embodiment,the map terrain data is a digital elevation model, or a portion of adigital elevation model.

The rendering module 280 identifies the one or more edges of thegeometric base that each connects two vertices of the set of verticesand intersects the tile border, as mentioned above. In an embodiment,the rendering module 280 identifies edges with end points (vertices)having respective coordinate locations in different map tiles. This maybe done by the rendering module 280, for example, by identifying, foreach vertex in the set of vertices, a map tile within which the vertexis located, then determining, for each edge connected to the vertex,whether a corresponding vertex at the other end of the edge is withinthe same map tile. If not, the rendering module 280 identifies the edgeas intersecting the tile border.

The rendering module 280 determines a set of sample points for the mapfeature based on the one or more edges that intersect the tile border.The set of sample points includes one or more candidate points that therendering module 280 uses to determine a sample elevation value to treatas the elevation value of the map feature as a whole. Selection ofsample points for the set of sample points can vary depending on theembodiment. In one embodiment, the rendering module 280 selects allvertices in the set of vertices as sample points. In a secondembodiment, the rendering module 280 selects a randomized subset ofvertices in the set of vertices as sample points.

In a third embodiment, the rendering module 280 selects as sample pointstwo projected span sample points, which is illustrated at FIG. 4A anddescribed in further detail below. In this case, the rendering module280 projects the edges of the map feature onto the tile border andidentifies a span encompassing the projected edges. The endpoints of thespan are the two projected span sample points. In an embodiment, therendering module 280 only projects edges that intersect the tile borderonto the tile border. Alternatively or additionally, the renderingmodule 280 identifies a vertex in the set of vertices that is a greatestdistance (e.g., Euclidian distance) from the tile border, and selectsthe vertex as a sample point.

The rendering module 280 determines a sample elevation value for eachsample point using the map terrain data. The rendering module 280determines, for each sample point in the set of sample points, arespective coordinate location, then determines the elevation value inthe map terrain data for the respective coordinate location as thesample elevation value for the sample point. The rendering module 280then selects a particular sample elevation value from the determinedsample elevation values. The selected particular sample elevation valueis used by the rendering module 280 as a baseline elevation value forthe map feature as a whole, e.g., as the elevation from which todetermine a height of the top surface graphical element, as detailedbelow. In an embodiment, the rendering module 280 selects as theparticular sample elevation value the sample elevation value with themaximum value. For example, if one sample point has an elevation valueof −15, a second sample point has an elevation value of 10, and a thirdsample point has an elevation value of 6, the rendering module 280selects the second sample point as the particular sample point.

The rendering module 280 determines vertex elevation values for eachvertex in the set of vertices based on the particular sample elevationvalue. For each vertex in the set of vertices, the rendering module 280determines an elevation of the vertex. For example, the rendering module280 identifies a respective coordinate location of each vertex, thendetermines the elevation value in the map terrain data for therespective coordinate location as the vertex elevation value for thevertex.

In an embodiment, as described further below with reference to FIG. 5 ,the rendering module 280 determines the vertex elevation valuesaccording to the following technique, which may be partially or whollyon the GPU 290. The rendering module 280 identifies a center point ofthe geometric base and a respective elevation using the map terraindata. The rendering module 280 generates an estimated slope for thegeometric base. For example, the rendering module 280 determines a slopefor each edge that intersects the tile border (e.g., the elevation ofthe first vertex connecting an edge minus the elevation of the secondvertex connecting the edge divided by a length of the edge). Therendering module 280 then averages each determined slope to generate theestimated slope. Based on the estimated slope and the center point, therendering module 280 determines a building base elevation. For example,the rendering module 280 starts at the center point and uses the slopeto calculate an elevation value estimate for a vertex that is furthestfrom the tile border. For example, if the center point is 5 distanceunits from the tile border and has an elevation value of 2, a mostdistant vertex is 10 distance units from the tile border, and theestimated slope is 1 elevation unit per distance unit, then therendering module 280 generates an elevation value estimate of 7 for themost distant vertex, which is used as the particular sample pointelevation for calculating vertex elevation values.

In an embodiment, the rendering module 280 determines sample elevationvalues, selects a particular sample elevation value, and/or determinesvertex elevation values only responsive to receiving each of the firstmap tile, the second map tile, and the map terrain data. In such anembodiment, the rendering module 280 may not render map featuresoverlapping both map tiles until both map tiles are received (e.g., thedigital map will not include a partial building on one of the map tilesbefore the other is loaded). Alternatively, the rendering module 280 mayrender a partial graphical element for a map feature overlapping two maptiles, then re-render the full graphical element upon receipt ofadditional data (e.g., the other map tile). In an embodiment, therendering module 280 waits until receiving both map tiles to render somegraphical elements, but renders partial graphical elements for othergraphical elements, depending upon the map features represented by thegraphical elements. For example, based on user input received at theclient device 120, or according to a style set by user input or receivedfrom the mapping system 110, the rendering module 280 renders partialgraphical elements for map features of particular map feature categories(e.g., rivers), but waits for both map tiles for map features ofdifferent map feature categories (e.g., buildings).

The rendering module 280 renders the first and second map tiles with themap feature using the vertex elevation values, but basing an elevationof a top surface graphical component of a graphical element representingthe map feature off the particular sample elevation value. The renderingmodule 280 determines an elevation of the top surface graphicalcomponent by adding the height value of the map feature to theparticular sample elevation value. For example, if the map feature is abuilding with a height value of 30 feet, and the particular sampleelevation value is 50 feet, the top surface graphical component is drawnby the rendering module 280 at an elevation of 80 feet from the baselineof the digital map. In an embodiment, the top surface graphicalcomponent is a flat surface with the same area as, and whollyoverlapping, the geometric base of the map feature. In otherembodiments, the map feature may include data specifying a particularshape of the top surface graphical component. In such embodiments, therendering module 280 draws the vertical edges to the top surfacegraphical component based on the data.

The rendering module 280 renders one or more side surface graphicalcomponents of the graphical element representing the map feature usingthe elevation of the top surface graphical component and the vertexelevation values of one or more of the vertices in the set of vertices.For each of the one or more vertices, the rendering module 280 draws avertical edge at the location of the vertex, from the vertex elevationvalue to the elevation of the top surface graphical component. For oneor more polygons generated upon the rendering module 280 drawing thevertical edges, the rendering module 280 renders a side surfacegraphical component. In some embodiments, the map feature includes dataspecifying polygons for which to not generate a side surface (e.g., athree-sided alcove with an open face).

For example, a map feature may have a geometric base with threevertices. A first vertex has a vertex elevation value of 5, a secondvertex has a vertex elevation value of 10, and a third vertex has avertex elevation value of 15. The map feature has a height value of 20.The rendering module 280 determines a particular sample elevation valueof 6. As such, the map feature has a top surface graphical componentelevation of 26. The rendering module 280 draws a vertical edge at thefirst vertex from an elevation of 5 to an elevation of 26, at the secondvertex from an elevation of 10 to an elevation of 26, and at the thirdvertex from an elevation of 15 to an elevation of 26. (Elevations may beor be based on, for example, a particular distance in map tile space,such as 1/64 of a map tile length, or a particular distance ingeographic space, such as feet, meters, yards, etc.). The renderingmodule 280 generates three side surfaces, one between the first andsecond vertices, one between the second and third vertices, and onebetween the first and third vertices.

The rendering module 280 displays the graphical element generated inthis manner on an electronic display of the client device 120. Forexample, the rendering module 280 generates the top surface graphicalcomponent and other graphical components of the graphical element forthe map feature based on data of the map feature, then renders thegraphical element as part of a digital map displayed at the electronicdisplay (which includes some or all of the first map tile and the secondmap tile).

After rendering the digital map including the graphical element, theclient device 120 may receive updated map data including updates to oneor more of the first map tile, second map tile, map feature, and mapterrain data. In such a situation, the rendering module 280 determinesan updated sample elevation value for an updated sample point of the setof sample points using the updated map data, determines updated vertexelevation values for the set of vertices using the updated sampleelevation value, and re-renders the digital map with the top surfacegraphical component having an elevation corresponding to a sum of theupdated sample elevation value and the height value. The renderingmodule 280 displays the re-rendered digital map.

In an embodiment, upon determining the vertex elevation values, therendering module 280 stores the vertex elevation values in a vertexshader of the GPU 290. The client device 120 uses the GPU 290 to renderthe map feature based off the stored vertex elevation values in thevertex shader by according to some or all of the techniques describedherein with reference to the rendering module 280, which, depending uponthe embodiment, may partially or wholly operate using the CPU 260.

If the rendering module 280 determines, for an additional map featurehaving an additional geometric base and an additional height value (thegeometric base represented by an additional set of vertices), that theadditional map feature resides entirely upon the first map tile, therendering module 280 may employ efficient rooftop rendering as describedbelow. The rendering module 280 may determine, using the map terraindata, an additional elevation value for a default point of theadditional geometric base. The rendering module 280 may determine thedefault point as a center point of the geometric base, which may be apoint of greatest distance from each vertex in the additional set ofvertices.

The rendering module 280 may then determine additional vertex elevationvalues for the additional set of vertices using the additional sampleelevation value and the additional height value and render the first maptile with the additional map feature using the additional vertexelevation values. The additional map feature may then be rendered by therendering module 280 on the first map tile with an additional topsurface graphical component having an elevation corresponding to a sumof the additional sample elevation value and the additional heightvalue.

Although described herein with reference to determining a top surfacegraphical component, the techniques described can be applied to anyalternative use cases where knowledge of full map feature geometry isneeded and could change over time, such as highlighting selected mapfeatures, calculating the area of a map feature, or so on.

The rendering module 280 also performs efficient rooftop rendering inconjunction with the GPU 290. By performing an efficient rooftoprendering technique, the rendering module 280 avoids creating acomputational bottleneck for the CPU 260 when rendering top surfacegraphical components, while simultaneously enabling asynchronousuploading of data to the GPU 290 for use in rendering the top surfacegraphical components (rather than requiring the gathering of all databefore initiating a rendering process for a map feature).

The rendering module 280 receives from a mapping server a map tilerepresenting a geographic region. The map tile is associated with a mapfeature having a geometric base and a height value, and the geometricbase is represented by a set of vertices on the map tile. The renderingmodule 280 also receives map terrain data including elevation values forpoints on the map tile.

The rendering module 280 determines whether all vertices of the mapfeature are in the set of vertices on the map tile. If they are not, therendering module 280 ends efficient rooftop rendering and schedules themap feature for rooftop reconstruction. If they are, then responsive todetermining all vertices of the map feature are in the set of verticeson the map tile, the rendering module 280 renders the map tile using theGPU 290.

The rendering module 280 uses the GPU 290 and/or the CPU 260 to select asample point on the map tile using the set of vertices. In particular,the rendering module 280 uses the GPU 290 to select as the sample pointa center point of the geometric base of the map feature.

The rendering module 280 uses the GPU 290 to determine a sampleelevation value for the sample point using the map terrain data. Forexample, the rendering module 280 uses the GPU 290 to determine acoordinate location of the sample point and retrieve an elevation valuefrom the map terrain data for the coordinate location.

The rendering module 280 uses the GPU 290 to determine vertex elevationvalues for respective vertices in the set of vertices using the sampleelevation value. In an embodiment, the rendering module 280 generates anestimated average slope of the map feature. This may involve determiningthe slope of one or more edges and averaging the slopes of the one ormore edges, depending upon the embodiment. The rendering module 280 usesthe estimated slope to generate a sample point elevation value for thesample point, and uses the sample point elevation value to determine thevertex elevation values.

In an embodiment, the GPU 290 determines, using the sample elevationvalue, an estimated slope for an area on the map tile encompassed by thegeometric base. For each vertex, to determine its elevation value, theGPU 290 determines a position of the vertex relative to the sample pointcorresponding to the sample elevation value and then determines a vertexelevation value for the vertex based on a change in elevation from thesample point to the determined position of the vertex according to theestimated slope. In an embodiment, the GPU 290 is configured to renderthe map tile using a three-dimensional shader language.

The rendering module 280 uses the GPU 290 to render the map tile withthe map feature using the vertex elevation values. The map feature isrendered in three dimensions with a top surface graphical component of agraphical element of the map feature having an elevation correspondingto a sum of the sample elevation value and the height value. Therendering module 280 displays the rendered map tile on an electronicdisplay of the client device.

III. Rooftop Reconstruction Visualization

FIG. 3 illustrates a three-dimensional digital map 300 including abuilding graphical element with a misaligned roof, according to anembodiment. In particular, the figure is a side view of a portion of arendered three-dimensional digital map 300 with graphical elementsrepresenting map features.

As described above, map features residing partially in multiple maptiles can lead to situations where the different portions of the mapfeature are rendered differently, e.g., with different top surfacegraphical component elevations, if not for the benefits of techniques ofthe present disclosure. The figure illustrates an embodiment of twopartial building footprints rendered separately by the client device120, and therefore with different top surface graphical components, aswould occur if the rendering module 280 did not implement the techniquesdescribed herein to render the building.

In the embodiment of the figure, map tile 310A borders map tile 310B attile border 320. The digital map 300 includes a map terrain 330 (e.g.,terrain data such as a digital elevation model) that specifies elevationvalues for points on the map tiles 310. In particular, the map tiles 310together represent a rolling hill with a building on top, with adescending slope of the hill intersecting the tile border 320 from thedirection of map tile 310A to map tile 310B. As such, the elevations ofpoints within the partial footprint of the building on map tile 310A aregreater than the elevations of points within the partial footprint ofthe building on map tile 310B.

The rendering module 280 determines a base elevation 350 for eachpartial building footprint. Due to the elevation differences in eachpartial building footprint, the partial building footprint in map tile310A leads to a respective base elevation 350A that is greater than baseelevation 350B of the partial building footprint in map tile 310B. Assuch, when a building height value 340 of the building is added by therendering module 280 to each base elevation 350 to determine arespective rooftop elevation 360 for each partial building, theresulting rooftop elevation 360A on map tile 310A is greater than theresulting rooftop elevation 360B on map tile 310B. As such, each partialbuilding is rendered by the rendering module 280 with a differentrooftop (at the respective rooftop elevation 360), leading to agraphical element representing the building that has an unrealistic roofat an incorrect height. Not only does the generated graphical elementincorrectly visualize the map feature, but it involved rooftop elevationcalculations for two partial building footprints, rather than onerooftop elevation calculation for one building footprint. As describedherein, the digital map can be improved and its generation can be madeless computationally expensive using rooftop reconstruction andefficient rooftop rendering techniques.

FIG. 4A illustrates a top-down topographic view of calculating a samplepoint for rooftop reconstruction in a digital map 400, according to anembodiment. The digital map 400 includes a map tile 405A and a map tile405B that meet at tile border 410. The client device 120 has map terraindata for the tiles, which is visualized in the figure as topographic mapelevation data 415. A building map feature's geometric base isillustrated as building feature base 420. The building feature base 420is a square with four vertices and four edges among the vertices. Thebuilding's geometric base overlaps both map tiles 405, particularly attwo edges that connect two vertices on the first map tile 405A to avertex on the second map tile 405B.

The rendering module 280 determines three sample points for the mapfeature. For two of the three, the rendering module 280 projects theedges onto the tile border 410 and identifies a border span 425 for themap feature. The rendering module 280 identifies as two sample pointsthe endpoints of the span of the projection of the building's edges ontothe tile border 410—in other words, the projected span sample points430. For the third sample point, the rendering module 280 identifies avertex that is a maximum distance (e.g., offset in coordinate space)from the tile border 410. This maximum border offset sample point 435for the building map feature is the third sample point.

FIG. 4B illustrates a side view of rooftop reconstruction in a digitalmap 460, according to one embodiment. The figure illustrates anotherembodiment of the rendered building and hill of FIG. 3 , this time withthe techniques described herein applied for rooftop reconstruction andefficient rooftop rendering. In contrast to the misaligned rooftopillustrated in FIG. 3 , the rooftop in FIG. 4B accurately visualizes theroof of the building in relation to the terrain on which the building issituated. FIG. 4B includes the map tile 310A, map tile 310B, and tileborder 320, along with the same map terrain 330.

Using rooftop reconstruction techniques, the rendering module 280matches the partial building footprints and generates one base elevation440 for the building footprint. The rendering module 280 adds thebuilding height value 445 to the base elevation 440 to determine arooftop elevation 450 for the building. The rendering module 280generates a top surface graphical component for a graphical elementrepresenting the building, e.g., a roof, which is not disjointed at thetile border 320 and which is properly elevated above the map terrain330.

IV. Efficient Graphical Element Rooftop Rendering Visualization

FIG. 5 illustrates a top-down topographic view of calculating a samplepoint for efficient rooftop rendering in a digital map, according to anembodiment. A map tile 510 includes map elevation data 520, atopographic illustration of map terrain data. The map tile 510 alsoincludes a building map feature with a building feature base 530, thegeometric base of the building. The building feature base sample pointfor GPU elevation estimation 540 (the center point 540 lessspecifically), is the center of the building.

The rendering module 280 uses the GPU 290 to determine the center point540 (or, if the map feature includes an indication of the center point,fetches the center point) and uses the map elevation data 520 todetermine an elevation of the center point 540. The GPU 290 estimates anaverage slope for the building. Using the estimated average slope andthe center point 540, the rendering module 280 estimates an elevation ofthe base of the building from which to determine the top surfacegraphical component elevation (effectively, a particular sampleelevation value).

V. Exemplary Processes

FIG. 6 is a flowchart 600 illustrating a process for rooftopreconstruction, according to one embodiment. In the embodiment shown,the steps of FIG. 6 are illustrated from the perspective of the clientdevice 120. In other embodiments, some or all of the steps may beperformed by other entities or components, such as the mapping system110. In addition, some embodiments may perform the steps in parallel,perform the steps in different orders, or perform different steps.

The client device 120 receives 610, from the mapping system 110, a firstmap tile representing a first portion of a geographic region. The firstmap tile is associated with a map feature that has a geometric base(e.g., footprint) and a height value. The geometric base is representedby a set of vertices that includes one or more vertices on the first maptile. The client device 120 also receives 610 a second map tilerepresenting a second portion of the geographic region. The set ofvertices includes one or more vertices on the second map tile. Theclient device 120 also receives 610 map terrain data including elevationvalues for points on the first map tile and points on the second maptile.

The client device 120 identifies 620, based on the set of vertices, oneor more edges of the geometric base that each connects two vertices ofthe set of vertices and intersects a tile border between the first maptile and the second map tile. For example, the client device 120identifies 620 an edge of the one or more edges with a first vertex inthe first map tile and a second vertex in the second map tile bycomparing coordinates of the first vertex and the second vertex tocoordinates describing respective geographic areas represented by thefirst map tile and the second map tile. If the first vertex is withinthe geographic area represented by the first map tile and the secondvertex is within the geographic area represented by the second map tile,the edge connecting the first vertex and the second vertex intersectsthe tile border.

Responsive to receiving each of the first map tile, the second map tile,and the terrain data 625, the client device 120 determines 630 a set ofsample points for the map feature based on the one or more edges thatintersect the tile border, selects 640 a particular sample elevationvalue from the determined sample elevation values, and determines 650vertex elevation values for respective vertices in the set of vertices.

The client device 120 renders 660 the first map tile and the second maptile with the map feature using the vertex elevation values. The mapfeature is rendered by the client device 120 (e.g., in three dimensions)with a top surface graphical component having an elevation correspondingto a sum of the particular sample elevation value and the height valueof the map feature.

The client device 120 displays 670 a graphical element of the mapfeature comprising the top surface graphical component on an electronicdisplay of the client device. For example, if the map feature is abuilding, the client device 120 displays 670 a digital map including thebuilding, as represented by a graphical element based on the geometricbase and topped by the top surface graphical component at the elevationdetermined as a sum of the particular sample elevation value and heightvalue.

In this manner, the client device 120 not only halves the computationalload on the processor used to generate the top surface graphicalcomponent of the building, but also creates a more representativegraphical element that more accurately visualizes the map feature withinthe digital map.

In an embodiment, the client device 120 determines the set of samplepoints by determining, based on the identified one or more edges, aborder intersection span for the map feature, the border intersectionspan representing a projection of the one or more edges onto the border,and selecting one or more end points of the border intersection span forinclusion in the set of sample points.

In an embodiment, the client device 120 determines the set of samplepoints by determining a vertex of the set of vertices having a maximumdistance from the border relative to the other vertices of the set ofvertices and including the vertex as a sample point in the set of samplepoints.

In an embodiment, the client device 120 receives the one or more of thefirst map tile, the second map tile, and the terrain dataasynchronously.

In an embodiment, the client device 120 determines the particular sampleelevation value by selecting the maximum sample elevation value from thesample elevation values for the set of sample points.

In an embodiment, after rendering the first and second map tiles, theclient device 120 receives updated map data including updates to one ormore of the first map tile, the second map tile, the map feature, andthe map terrain data. The client device 120 determines an updated sampleelevation value for an updated sample point of the set of sample pointsusing the updated map data. The client device 120 determining updatedvertex elevation values for the set of vertices using the updated sampleelevation value. The client device 120 re-renders the first and secondmap tiles with the map feature using the updated vertex elevationvalues. The map feature is rendered with the top surface graphicalcomponent having an elevation corresponding to a sum of the updatedsample elevation value and the height value. The client device 120displays the re-rendered first and second map tiles on the electronicdisplay of the client device.

In an embodiment, the client device 120 determines the vertex elevationvalues by, responsive to receiving the map terrain data, evaluating eachmap tile of the first and second map tiles for map features intersectingthe tile border of the first map tile and second map tile. This caninvolve the client device 120 identifying a first portion of the mapfeature on the first map tile intersecting the tile border between thefirst and second map tiles. The client device 120 then compares thefirst portion of the map feature to one or more map features of thesecond map tile. The one or more map features of the second map tileinclude a second portion of the map feature. The client device 120 thendetermines, based on the comparison, that that first portion of the mapfeature matches the second portion of the map feature of the second maptile. The client device 120, responsive to determining that the firstand second portions match, determines the vertex elevation values forthe set of vertices based on the matching first and second portions ofthe map feature. Depending upon the embodiment, this may furtherinvolve, in a case where the first map tile is received before thesecond map tile, and before receiving the second map tile, responsive todetermining that one or more vertices of the set of vertices are not onthe first map tile, rendering, by the client device 120, the first maptile without the map feature. Determining the vertex elevation valuesmay, in an embodiment, involve, before evaluating each map tile of thefirst and second map tiles for map features intersecting the tileborders of the first and second map tiles, ordering, by the clientdevice 120, one or more map features of the first map tile thatintersect a first tile border of the first map tile according to a pointof intersection of the one or map features on the first tile border. Theclient device 120 then orders one or more map features of the second maptile that intersect a second tile border of the second map tileaccording to a point of intersection of the one or map features on thesecond tile border.

In an embodiment, the client device 120 stores the vertex elevationvalues in a vertex shader, and the first and second map tiles arerendered with the map feature by the GPU 290 of the client device 120using the vertex shader.

FIG. 7 is a flowchart 700 illustrating a process for efficient rooftoprendering, according to one embodiment. In the embodiment shown, thesteps of FIG. 7 are illustrated from the perspective of the clientdevice 120. In other embodiments, some or all of the steps may beperformed by other entities or components, such as the mapping system110. In addition, some embodiments may perform the steps in parallel,perform the steps in different orders, or perform different steps.

The client device 120 receives 710, from the mapping system 110, a maptile representing a geographic region. The map tile is associated with amap feature having a geometric base and a height value, and thegeometric base is represented by a set of vertices on the map tile. Theclient device 120 also receives 720, from the mapping system 110, mapterrain data including elevation values for points on the map tile. Theclient device 120 determines 730 whether all vertices of the map featureare in the set of vertices on the map tile.

The client device 120, responsive to determining all vertices of the mapfeature are in the set of vertices on the map tile, renders 735 the maptile using the GPU 290, the rendering 735 comprising the followingsteps. The GPU 290 of the client device 120 selects 740 a sample pointon the map tile using the set of vertices. The GPU 290 of the clientdevice 120 determines 750 a sample elevation value for the sample pointusing the map terrain data. The GPU 290 of the client device 120determines 760 vertex elevation values for respective vertices in theset of vertices using the sample elevation value. The GPU 290 of theclient device 120 renders 770 the map tile with the map feature usingthe vertex elevation values. The map feature is rendered by the GPU 290of the client device 120 in three dimensions with a top surfacegraphical component of a graphical element of the map feature having anelevation corresponding to a sum of the sample elevation value and theheight value.

The client device 120 displays 780 the rendered first and second maptiles on an electronic display of the client device 120.

In this manner, the client device 120 takes advantage of the GPU 290 toreduce the computational expense in rendering graphical elements in adigital map. Rather than determining the elevation for each point withinthe geometric base of the map feature on the CPU 260, the client device120 utilizes the GPU 290 to determine a base elevation based off onesample point (e.g., a center point), and an average slope (e.g., anaverage of respective slope values of vertices in the geometric base, asindicated by the terrain data). What's more, the technique does notrequire preparing information about building elevation beforehand, andenables asynchronous upload of split data (e.g., map feature data andterrain data) to the GPU 290 where it is combined to generate thegraphical element.

In an embodiment, the selected sample point corresponds to a centerpoint of the geometric base of the map feature. In an embodiment, theclient device receives the map tile and the terrain data asynchronously.In an embodiment, the map feature is a building, and the top surfacegraphical component represents a rooftop of the building. In anembodiment, the GPU is configured to render the map tile using athree-dimensional shader language.

In an embodiment, the GPU 290 of the client device 120 determining thevertex elevation values includes the following steps. The GPU 290 of theclient device 120 determines, using the sample elevation value, anestimated slope for an area on the map tile encompassed by the geometricbase. For each vertex, the GPU 290 of the client device 120 determines aposition of the vertex relative to the sample point corresponding to thesample elevation value and then a vertex elevation value for the vertexbased on a change in elevation from the sample point to the determinedposition of the vertex according to the estimated slope.

In an embodiment, the client device 120, after rendering the map tile,receives updated map data including updates to one or more of the maptile, the map feature, or the map terrain data. The GPU 290 of theclient device 120 determines an updated sample elevation value for anupdated sample point of the set of sample points using the updated mapdata. The GPU 290 of the client device 120 determines updated vertexelevation values for the set of vertices using the updated sampleelevation value. The GPU 290 of the client device 120 re-renders the maptile with the map feature using the updated vertex elevation values,wherein the map feature is rendered with the top surface graphicalcomponent having an elevation corresponding to a sum of the updatedsample elevation value and the height value. The client device 120displays the re-rendered map tile on the electronic display of theclient device 120.

In an embodiment, the client device 120 determines whether all verticesof a second map feature are in a second set of vertices on a particularmap tile. The client device 120, responsive to determining that not allvertices of the second map feature are in the second set of vertices,performs rooftop reconstruction for the second map feature. Responsiveto performing rooftop reconstruction for the second map feature, theclient device 120 renders the map tile using the GPU 290.

VI. Computer Diagram

FIG. 8 is a block diagram that illustrates a computer system 800 uponwhich embodiments of components of the computing environment 100 may beimplemented. Computer system 800 includes a bus 802 or othercommunication mechanism for communicating information, and a hardwareprocessor 804 coupled with bus 802 for processing information. Hardwareprocessor 804 may be, for example, a general purpose microprocessor.

Example computer system 800 also includes a main memory 806, such as arandom access memory (RAM) or other dynamic storage device, coupled tobus 802 for storing information and instructions to be executed byprocessor 804. Main memory 806 also may be used for storing temporaryvariables or other intermediate information during execution ofinstructions to be executed by processor 804. Such instructions, whenstored in non-transitory storage media accessible to processor 804,render computer system 800 into a special-purpose machine that iscustomized to perform the operations specified in the instructions.

Computer system 800 further includes a read only memory (ROM) 808 orother static storage device coupled to bus 802 for storing staticinformation and instructions for processor 804. A storage device 810,such as a magnetic disk or optical disk, is provided and coupled to bus802 for storing information and instructions.

Computer system 800 may be coupled via bus 802 to a display 812, such asa LCD screen, LED screen, or touch screen, for displaying information toa computer user. An input device 814, which may include alphanumeric andother keys, buttons, a mouse, a touchscreen, or other input elements iscoupled to bus 802 for communicating information and command selectionsto processor 804. In some embodiments, the computer system 800 may alsoinclude a cursor control 816, such as a mouse, a trackball, or cursordirection keys for communicating direction information and commandselections to processor 804 and for controlling cursor movement ondisplay 812. The cursor control 816 typically has two degrees of freedomin two axes, a first axis (e.g., x) and a second axis (e.g., y), thatallows the device to specify positions in a plane.

Computer system 800 may implement the techniques described herein usingcustomized hard-wired logic, one or more ASICs or FPGAs, firmware andprogram logic which in combination with the computer system causes orprograms computer system 800 to be a special-purpose machine. Accordingto one embodiment, the techniques herein are performed by computersystem 800 in response to processor 804 executing one or more sequencesof one or more instructions contained in main memory 806. Suchinstructions may be read into main memory 806 from another storagemedium, such as storage device 810. Execution of the sequences ofinstructions contained in main memory 806 causes processor 804 toperform the process steps described herein. In alternative embodiments,hard-wired circuitry may be used in place of or in combination withsoftware instructions.

The term “storage media” as used herein refers to any non-transitorymedia that store data and instructions that cause a machine to operationin a specific fashion. Such storage media may comprise non-volatilemedia and volatile media. Non-volatile media includes, for example,optical or magnetic disks, such as storage device 810. Volatile mediaincludes dynamic memory, such as main memory 806. Common forms ofstorage media include, for example, a floppy disk, a flexible disk, harddisk, solid state drive, magnetic tape, or any other magnetic datastorage medium, a CD-ROM, any other optical data storage medium, anyphysical medium with patterns of holes, a RAM, a PROM, and EPROM, aFLASH-EPROM, NVRAM, any other memory chip or cartridge.

Storage media is distinct from but may be used in conjunction withtransmission media. Transmission media participates in transferringinformation between storage media. For example, transmission mediaincludes coaxial cables, copper wire and fiber optics, including thewires that comprise bus 802. Transmission media can also take the formof acoustic, radio, or light waves, such as those generated duringradio-wave and infra-red data communications, such as WI-FI, 3G, 4G,BLUETOOTH, or wireless communications following any other wirelessnetworking standard.

Various forms of media may be involved in carrying one or more sequencesof one or more instructions to processor 804 for execution. For example,the instructions may initially be carried on a magnetic disk or solidstate drive of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 800 canreceive the data on the telephone line and use an infra-red transmitterto convert the data to an infra-red signal. An infra-red detector canreceive the data carried in the infra-red signal and appropriatecircuitry can place the data on bus 802. Bus 802 carries the data tomain memory 806, from which processor 804 retrieves and executes theinstructions. The instructions received by main memory 806 mayoptionally be stored on storage device 810 either before or afterexecution by processor 804.

Computer system 800 also includes a communication interface 818 coupledto bus 802. Communication interface 818 provides a two-way datacommunication coupling to a network link 820 that is connected to alocal network 822. For example, communication interface 818 may be anintegrated services digital network (ISDN) card, cable modem, satellitemodem, or a modem to provide a data communication connection to acorresponding type of telephone line. As another example, communicationinterface 818 may be a local area network (LAN) card to provide a datacommunication connection to a compatible LAN. Wireless links may also beimplemented. In any such implementation, communication interface 818sends and receives electrical, electromagnetic or optical signals thatcarry digital data streams representing various types of information.

Network link 820 typically provides data communication through one ormore networks to other data devices. For example, network link 820 mayprovide a connection through local network 822 to a host computer 824 orto data equipment operated by an Internet Service Provider (ISP) 826.ISP 826 in turn provides data communication services through the worldwide packet data communication network now commonly referred to as the“Internet” 828. Local network 822 and Internet 828 both use electrical,electromagnetic or optical signals that carry digital data streams. Thesignals through the various networks and the signals on network link 820and through communication interface 818, which carry the digital data toand from computer system 800, are example forms of transmission media.

Computer system 800 can send messages and receive data, includingprogram code, through the network(s), network link 820 and communicationinterface 818. In the Internet example, a server 830 might transmit arequested code for an application program through Internet 828, ISP 826,local network 822 and communication interface 818. The received code maybe executed by processor 804 as it is received, and stored in storagedevice 810, or other non-volatile storage for later execution.

VII. Additional Considerations

The foregoing description of the embodiments has been presented for thepurpose of illustration; it is not intended to be exhaustive or to limitthe patent rights to the precise forms disclosed. Persons skilled in therelevant art can appreciate that many modifications and variations arepossible in light of the above disclosure.

Some portions of this description describe the embodiments in terms ofalgorithms and symbolic representations of operations on information.These algorithmic descriptions and representations are commonly used bythose skilled in the data processing arts to convey the substance oftheir work effectively to others skilled in the art. These operations,while described functionally, computationally, or logically, areunderstood to be implemented by computer programs or equivalentelectrical circuits, microcode, or the like. Furthermore, it has alsoproven convenient at times, to refer to these arrangements of operationsas modules, without loss of generality. The described operations andtheir associated modules may be embodied in software, firmware,hardware, or any combinations thereof.

Any of the steps, operations, or processes described herein may beperformed or implemented with one or more hardware or software modules,alone or in combination with other devices. In one embodiment, asoftware module is implemented with a computer program productcomprising a computer-readable medium containing computer program code,which can be executed by a computer processor for performing any or allof the steps, operations, or processes described.

Embodiments may also relate to an apparatus for performing theoperations herein. This apparatus may be specially constructed for therequired purposes, and/or it may comprise a general-purpose computingdevice selectively activated or reconfigured by a computer programstored in the computer. Such a computer program may be stored in anon-transitory, tangible computer readable storage medium, or any typeof media suitable for storing electronic instructions, which may becoupled to a computer system bus. Furthermore, any computing systemsreferred to in the specification may include a single processor or maybe architectures employing multiple processor designs for increasedcomputing capability.

Embodiments may also relate to a product that is produced by a computingprocess described herein. Such a product may comprise informationresulting from a computing process, where the information is stored on anon-transitory, tangible computer readable storage medium and mayinclude any embodiment of a computer program product or other datacombination described herein.

As used herein, any reference to “one embodiment” or “an embodiment”means that a particular element, label, structure, or characteristicdescribed in connection with the embodiment is included in at least oneembodiment. The appearances of the phrase “in one embodiment” in variousplaces in the specification are not necessarily all referring to thesame embodiment. Similarly, use of “a” or “an” preceding an element orcomponent is done merely for convenience. This description should beunderstood to mean that one or more of the element or component ispresent unless it is obvious that it is meant otherwise.

Where values are described as “approximate” or “substantially” (or theirderivatives), such values should be construed as accurate +/−10% unlessanother meaning is apparent from the context. From example,“approximately ten” should be understood to mean “in a range from nineto eleven.”

As used herein, the terms “comprises,” “comprising,” “includes,”“including,” “has,” “having” or any other variation thereof, areintended to cover a non-exclusive inclusion. For example, a process,method, article, or apparatus that comprises a list of elements is notnecessarily limited to only those elements but may include otherelements not expressly listed or inherent to such process, method,article, or apparatus. Further, unless expressly stated to the contrary,“or” refers to an inclusive or and not to an exclusive or. For example,a condition A or B is satisfied by any one of the following: A is true(or present) and B is false (or not present), A is false (or notpresent) and B is true (or present), and both A and B are true (orpresent).

Upon reading this disclosure, those of skill in the art will appreciatestill additional alternative structural and functional designs that maybe used to employ the described techniques and approaches. Thus, whileparticular embodiments and applications have been illustrated anddescribed, it is to be understood that the described subject matter isnot limited to the precise construction and components disclosed. Thescope of protection should be limited only by the following claims.

Finally, the language used in the specification has been principallyselected for readability and instructional purposes, and it may not havebeen selected to delineate or circumscribe the patent rights. It istherefore intended that the scope of the patent rights be limited not bythis detailed description, but rather by any claims that issue on anapplication based hereon. Accordingly, the disclosure of the embodimentsis intended to be illustrative, but not limiting, of the scope of thepatent rights, which is set forth in the following claims.

What is claimed is:
 1. A method comprising: receiving, by a clientdevice from a mapping system: a first map tile representing a firstportion of a geographic region, the first map tile associated with a mapfeature having a geometric base and a height value, wherein thegeometric base is represented by a set of vertices that includes one ormore vertices on the first map tile, a second map tile representing asecond portion of the geographic region, wherein the set of verticesincludes one or more vertices on the second map tile, and map terraindata including elevation values for points on the first map tile andpoints on the second map tile; identifying, based on the set ofvertices, one or more edges of the geometric base that connects twovertices of the set of vertices and intersects a tile border between thefirst map tile and the second map tile; determining a set of samplepoints for the map feature based on the one or more edges that intersectthe tile border; responsive to receiving each of the first map tile, thesecond map tile, and the terrain data: determining a sample elevationvalue for each sample point of the set of sample points using the mapterrain data; selecting a particular sample elevation value from thedetermined sample elevation values; and determining vertex elevationvalues for respective vertices in the set of vertices; rendering thefirst and second map tiles with the map feature using the vertexelevation values, wherein the map feature is rendered in threedimensions with a top surface graphical component having an elevationcorresponding to a sum of the particular sample elevation value and theheight value; and displaying a graphical element of the map featurecomprising the top surface graphical component on an electronic displayof the client device.
 2. The method of claim 1, wherein determining theset of sample points comprises: determining, based on the identified oneor more edges, a border intersection span for the map feature, theborder intersection span representing a projection of the one or moreedges onto the border; and selecting one or more end points of theborder intersection span for inclusion in the set of sample points. 3.The method of claim 1, wherein determining the set of sample pointscomprises: determining a vertex of the set of vertices having a maximumdistance from the border relative to the other vertices of the set ofvertices; and including the vertex as a sample point in the set ofsample points.
 4. The method of claim 1, wherein the client devicereceives the one or more of the first map tile, the second map tile, andthe terrain data asynchronously.
 5. The method of claim 1, whereindetermining the particular sample elevation value comprises: selectingthe maximum sample elevation value from the sample elevation values forthe set of sample points.
 6. The method of claim 1, further comprising:after rendering the first and second map tiles, receiving, by the clientdevice, updated map data including updates to one or more of the firstmap tile, the second map tile, the map feature, and the map terraindata; determining an updated sample elevation value for an updatedsample point of the set of sample points using the updated map data;determining updated vertex elevation values for the set of verticesusing the updated sample elevation value; re-rendering the first andsecond map tiles with the map feature using the updated vertex elevationvalues, wherein the map feature is rendered with the top surfacegraphical component having an elevation corresponding to a sum of theupdated sample elevation value and the height value; and displaying there-rendered first and second map tiles on the electronic display of theclient device.
 7. The method of claim 1, wherein determining the vertexelevation values comprises: responsive to receiving the map terraindata, evaluating each map tile of the first and second map tiles for mapfeatures intersecting the tile border of the first map tile and secondmap tile, comprising: identifying a first portion of the map feature onthe first map tile intersecting the tile border between the first andsecond map tiles; comparing the first portion of the map feature to oneor more map features of the second map tile, the one or more mapfeatures of the second map tile including a second portion of the mapfeature; determining, based on the comparison, that that first portionof the map feature matches the second portion of the map feature of thesecond map tile; and responsive to determining that the first and secondportions match, determining the vertex elevation values for the set ofvertices based on the matching first and second portions of the mapfeature.
 8. The method of claim 7, wherein the first map tile isreceived before the second map tile, and wherein the evaluating furthercomprises: before receiving the second map tile, responsive todetermining that one or more vertices of the set of vertices are not onthe first map tile, rendering the first map tile without the mapfeature.
 9. The method of claim 7, further comprising: before evaluatingeach map tile of the first and second map tiles for map featuresintersecting the tile borders of the first and second map tiles,ordering one or more map features of the first map tile that intersect afirst tile border of the first map tile according to a point ofintersection of the one or map features on the first tile border, andordering one or more map features of the second map tile that intersecta second tile border of the second map tile according to a point ofintersection of the one or map features on the second tile border. 10.The method of claim 1, wherein the client device stores the vertexelevation values in a vertex shader, and wherein the first and secondmap tiles are rendered with the map feature by a graphics processingunit (GPU) of the client device using the vertex shader.
 11. The methodof claim 1, wherein the first map tile is associated with an additionalmap feature having an additional geometric base and an additional heightvalue, the geometric base represented by an additional set of vertices,and wherein rendering the first map tile further comprises: responsiveto determining the additional set of vertices are entirely on the firstmap tile, determining, using the map terrain data, an additionalelevation value for a default point of the additional geometric base;determining additional vertex elevation values for the additional set ofvertices using the additional sample elevation value and the additionalheight value; and rendering the first map tile with the additional mapfeature using the additional vertex elevation values, wherein theadditional map feature is rendered on the first map tile with anadditional top surface graphical component having an elevationcorresponding to a sum of the additional sample elevation value and theadditional height value.
 12. The method of claim 1, wherein determiningthe vertex elevation values for the one or more vertices comprises:determining, using the particular sample elevation value, an estimatedslope for an area on the first map tile encompassed by the geometricbase; and for each vertex of the one or more vertices: determining aposition of the vertex relative to the sample point corresponding to theparticular sample elevation value; and determining a vertex elevationvalue for the vertex based on a change in elevation from the samplepoint to the determined position of the vertex according to theestimated slope.
 13. A non-transitory computer-readable storage mediumcomprising instructions that, when executed by a computer processor,cause the processor to perform operations, the instructions comprisinginstructions to: receive, by a client device from a mapping system, afirst map tile representing a first portion of a geographic region, thefirst map tile associated with a map feature having a geometric base anda height value, wherein the geometric base is represented by a set ofvertices that includes one or more vertices on the first map tile, asecond map tile representing a second portion of the geographic region,wherein the set of vertices includes one or more vertices on the secondmap tile, and map terrain data including elevation values for points onthe first map tile and points on the second map tile; identify, based onthe set of vertices, one or more edges of the geometric base thatconnects two vertices of the set of vertices and intersects a tileborder between the first map tile and the second map tile; determine aset of sample points for the map feature based on the one or more edgesthat intersect the tile border; responsive to receiving each of thefirst map tile, the second map tile, and the terrain data: determine asample elevation value for each sample point of the set of sample pointsusing the map terrain data; select a particular sample elevation valuefrom the determined sample elevation values; and determine vertexelevation values for respective vertices in the set of vertices; renderthe first and second map tiles with the map feature using the vertexelevation values, wherein the map feature is rendered in threedimensions with a top surface graphical component having an elevationcorresponding to a sum of the particular sample elevation value and theheight value; and display a graphical element of the map featurecomprising the top surface graphical component on an electronic displayof the client device.
 14. The non-transitory computer-readable storagemedium of claim 13, wherein determining the set of sample pointscomprises instructions to: determine, based on the identified one ormore edges, a border intersection span for the map feature, the borderintersection span representing a projection of the one or more edgesonto the border; and select one or more end points of the borderintersection span for inclusion in the set of sample points.
 15. Thenon-transitory computer-readable storage medium of claim 13, whereindetermining the set of sample points comprises instructions to:determine a vertex of the set of vertices having a maximum distance fromthe border relative to the other vertices of the set of vertices; andinclude the vertex as a sample point in the set of sample points. 16.The non-transitory computer-readable storage medium of claim 13, whereindetermining the particular sample elevation value comprises aninstruction to: select the maximum sample elevation value from thesample elevation values for the set of sample points.
 17. Thenon-transitory computer-readable storage medium of claim 13, whereindetermining the vertex elevation values comprises instructions to:responsive to receiving the map terrain data, evaluate each map tile ofthe first and second map tiles for map features intersecting the tileborder of the first map tile and second map tile, comprising: identify afirst portion of the map feature on the first map tile intersecting thetile border between the first and second map tiles; compare the firstportion of the map feature to one or more map features of the second maptile, the one or more map features of the second map tile including asecond portion of the map feature; determine, based on the comparison,that that first portion of the map feature matches the second portion ofthe map feature of the second map tile; and responsive to determiningthat the first and second portions match, determine the vertex elevationvalues for the set of vertices based on the matching first and secondportions of the map feature.
 18. A system, comprising: a computerprocessor; and a non-transitory computer-readable storage mediumcomprising instructions that, when executed by the computer processor,cause the processor to perform operations, the instructions comprisinginstructions to: receive, by a client device from a mapping system, afirst map tile representing a first portion of a geographic region, thefirst map tile associated with a map feature having a geometric base anda height value, wherein the geometric base is represented by a set ofvertices that includes one or more vertices on the first map tile, asecond map tile representing a second portion of the geographic region,wherein the set of vertices includes one or more vertices on the secondmap tile, and map terrain data including elevation values for points onthe first map tile and points on the second map tile; identify, based onthe set of vertices, one or more edges of the geometric base thatconnects two vertices of the set of vertices and intersects a tileborder between the first map tile and the second map tile; determine aset of sample points for the map feature based on the one or more edgesthat intersect the tile border; responsive to receiving each of thefirst map tile, the second map tile, and the terrain data: determine asample elevation value for each sample point of the set of sample pointsusing the map terrain data; select a particular sample elevation valuefrom the determined sample elevation values; and determine vertexelevation values for respective vertices in the set of vertices; renderthe first and second map tiles with the map feature using the vertexelevation values, wherein the map feature is rendered in threedimensions with a top surface graphical component having an elevationcorresponding to a sum of the particular sample elevation value and theheight value; and display a graphical element of the map featurecomprising the top surface graphical component on an electronic displayof the client device.
 19. The system of claim 18, wherein determiningthe set of sample points comprises instructions to: determine, based onthe identified one or more edges, a border intersection span for the mapfeature, the border intersection span representing a projection of theone or more edges onto the border; and select one or more end points ofthe border intersection span for inclusion in the set of sample points.20. The system of claim 18, wherein determining the set of sample pointscomprises instructions to: determine a vertex of the set of verticeshaving a maximum distance from the border relative to the other verticesof the set of vertices; and include the vertex as a sample point in theset of sample points.